International Journal of Fracture 126: 345–384, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. T-stress in orthotropic functionally graded materials: Lekhnitskii and Stroh formalisms JEONG-HO KIM and GLAUCIO H. PAULINO ∗ Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Laboratory, 205 North Mathews Avenue, Urbana, IL 61801, U.S.A. ∗ Corresponding Author (E-mail: paulino@uiuc.edu; Phone: (217)333-3817; Fax: (217)265-8041) Received 18 November 2002; accepted in revised form 13 January 2004 Abstract. A new interaction integral formulation is developed for evaluating the elastic T-stress for mixed-mode crack problems with arbitrarily oriented straight or curved cracks in orthotropic nonhomogeneous materials. The development includes both the Lekhnitskii and Stroh formalisms. The former is physical and relatively simple, and the latter is mathematically elegant. The gradation of orthotropic material properties is integrated into the ele- ment stiffness matrix using a “generalized isoparametric formulation” and (special) graded elements. The specific types of material gradation considered include exponential and hyperbolic-tangent functions, but micromechanics models can also be considered within the scope of the present formulation. This paper investigates several fracture problems to validate the proposed method and also provides numerical solutions, which can be used as benchmark results (e.g. investigation of fracture specimens). The accuracy of results is verified by comparison with analytical solutions. Key words: functionally graded material (FGM), fracture mechanics, orthotropic materials, T-stress, interaction integral, two-state integral, finite element method (FEM), generalized isoparametric formulation (GIF). 1. Introduction The non-singular stress (T-stress) of the Williams’s eigenfunction expansion (Williams, 1957) influences crack growth under mixed-mode loading (Williams and Ewing, 1972; Ueda et al., 1983; Smith et al., 2001). Williams and Ewing (1972), and Ueda et al. (1983) performed experiments on polymethyl-methacrylate (PMMA) with a slanted internal crack, and found that the elastic T-stress influences the crack initiation angle. Smith et al. (2001) revisited earlier experimental results for brittle fracture of PMMA, and re-examined the role of the T-stress in brittle materials. The T-stress also influences crack path stability for mode I loading with a small imperfection (Cotterell and Rice, 1980). The T-stress influences crack-tip constraint and toughness under plane strain conditions (see, for example, O’Dowd et al., 1995). Larsson and Carlson (1973) investigated the T-stress, and observed that it affects the size and shape of the plastic zone. Betegón and Hancock (1991) investigated the two-parameter (J -T ) characterization of elastic-plastic crack-tip fields. Du and Hancock (1991) investigated the effect of the T-stress on the small-scale yielding field in elastic perfectly-plastic materials. Furthermore, O’Dowd and Shih (1991) developed the J -Q theory (Q is a hydrostatic stress parameter) and found that the Q-family provides a framework for quantifying the evolution of constraint from small-scale yielding to full yielding condi- tions. They deduced a one-to-one correspondence between Q and T , which is valid in the case where the applied load and geometry affect Q only through T . O’Dowd and Shih (1992)